Vertex-degree parity in large implicit “exchange graphs” implies some EP theorems asserting the existence of a second object without evidently providing a polytime algorithm for finding a second object.

[5] Kathie Cameron, Krawczyk's graphs show Thomason's algorithm for finding a second Hamilton cycle through a given edge in a cubic graph is exponential, Fifth Czech-Slovak Symposium on Combinatorics, Graph Theory, Algorithms and Applications, Prague; SIAM Conference on Discrete Mathematics, Toronto; July 1998.